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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 6, JUNE 2011
On Transceiver Signal Linearization and the Decoding Delay of Maximum Rate Complex Orthogonal Space-Time Block Codes Sarah Spence Adams, Member, IEEE, Nathaniel Karst, Mathav Kishore Murugan, and Tadeusz A. Wysocki, Senior Member, IEEE
Abstract—Complex orthogonal designs (CODs) have been successfully implemented in wireless systems as complex orthogonal space-time block codes (COSTBCs). Certain properties of the underlying CODs affect the performance of the codes. In addition to the main properties of a COD’s rate and decoding delay, a third consideration is whether the COD can achieve transceiver signal linearization, a property that facilitates practical implementation by, for example, significantly simplifying the receiver structure for iterative decoding. It has been shown that a COD can achieve this transceiver signal linearization if the nonzero entries in any given row of the matrix are either all conjugated or all nonconjugated. This paper determines the conditions under which maximum rate CODs can achieve this desirable property. For an odd number of transmit antennas, it is shown that maximum rate CODs that achieve the lower bound on decoding delay can also achieve transceiver signal linearization. In contrast, for an even number of transmit antennas, maximum rate CODs that achieve the lower bound on delay cannot achieve this linearization. In this latter case, linearization is possible only if the COD achieves at least twice the lower bound on delay. This work highlights the tradeoffs among these three important properties. Index Terms—Complex orthogonal designs (CODs), iterative decoding, space-time block codes, transceiver signal linearization.
have been applied as complex orthogonal space-time block codes (COSTBCs), which are useful in wireless applications due to their simple maximum-likelihood decoding rule and their guarantee of full diversity [1]. Several factors must be considered when choosing the underlying COD for application in a practical coding system. The rate (i.e., the ratio of the number of variables to the number of rows) and the minimum decoding delay (i.e., the minimum number of rows) for a given rate are two of the fundamental considerations. A third consideration is whether the COD allows for a linearized description of the received signal, as achieved by the original Alamouti code [6]. In his fundamental paper, Alamouti [6] used the following underlying COD: (1) Alamouti showed that for a flat-fading channel that is assumed constant over two consecutive symbol times, the received signal for his transmit diversity system can be expressed as
I. INTRODUCTION
W
E define an complex orthogonal dematrix with ensign (COD) as an such that tries from , where is the Hermitian transpose and is the identity matrix [1], [2]. Geramita and Seberry provide a comprehensive review of classical orthogonal designs [3], and Liang reviews and defines their generalizations [4]. Examples of the type of CODs considered in this paper can be found readily in the literature (e.g., [2], [5]). Such CODs
Manuscript received July 07, 2009; revised June 30, 2010; accepted November 11, 2010. Date of current version May 25, 2011. This work was supported in part by NSA Young Investigator Awards H98230-07-1-0022 and H98230-10-1-0220 and in part by Olin College. S. S. Adams is with the Franklin W. Olin College of Engineering, Needham, MA 02492-1245 USA (e-mail:
[email protected]). N. Karst is with the Center for Applied Mathematics, Cornell University, Ithaca, NY 14853 USA (e-mail:
[email protected]). M. K. Murugan was with the Indian Institute of Technology, Kharagpur, India. He is now with the Center for Applied Mathematics, Cornell University, Ithaca, NY 14853 USA (e-mail:
[email protected]). T. A. Wysocki is with the Peter Kiewit Institute, University of Nebraska—Lincoln, NE 68508 USA (e-mail:
[email protected]). Communicated by B. S. Rajan, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2011.2137050
where and are the received signals at times and , respectively, and are complex receiver noises, and and are the complex channel gain coefficients. Therefore, the receiver can be based on the use of a linear combiner producing two combined signals
which are then sent to the maximum likelihood detector for decoupled decoding. The full concept of the process is described in detail in Haykin and Moher’s textbook [7]. Alamouti showed that this process can be easily extended to multiple receive antennas [6], and Lu and Wang further extended the approach to multiuser scenarios [8]. In general, the same receiver structure based on a linear combiner can be applied if the received signal vector has the following form:
where the elements of the vector are either samples of the matrix conreceived signal or their conjugates, the tains complex linear combinations of channel state coefficients and their conjugates, and the th component of represents the
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ADAMS et al.: MAXIMUM RATE COMPLEX ORTHOGONAL SPACE-TIME BLOCK CODES
equivalent noise effect observed in time slot , [9]. This latter form is often referred to in the literature (e.g., [8]–[11]) as a linearized description of the transceiver signal, and we say that the code achieves transceiver signal linearization. This description allows for the application of a linear combiner before a maximum likelihood detector. This significantly simplifies the receiver structure, particularly for iterative decoding such as in the receivers studied by Lu and Wang [8] and the RAKE-based receivers studied by Jayaweera and Poor [10]. Su, Batalama, and Pados recently determined the conditions under which a COSTBC permits such a linearized received signal expression [9]. Specifically, they showed that if the underlying COD can be arranged so that the nonzero entries in any given row are either all conjugated or all nonconjugated, can achieve this linearization [9]. If achieves this then property, we say that it is conjugation-separated. Su et al. also noted that the tranceiver signal linearization is important because it allows for backwards compatibility with a variety of signal processing techniques and for the design of certain low complexity filters and equalizers [9]. Su et al. focused on determining when square CODs can achieve conjugation-separation, hence transceiver signal linearization [9]. Unfortunately, they determined that the rate of such square CODs approaches zero as the number of columns (i.e., antennas) increases [9]. Since a higher rate is preferred in practice, their work motivated us to determine the conditions under which rectangular CODs of higher rates, namely of maximum rate, can achieve transceiver signal linearization. Liang or determined that the maximum rate for a COD with columns is [2], so our focus is to determine the conditions under which a COD with this maximum rate can achieve transceiver signal linearization. Since these maximum rate CODs are no longer square, as opposed to the square CODs considered by Su et al. [9], we must also aim to minimize their decoding delay. Adams et al.. determined that a lower bound or on decoding delay for maximum rate CODs with columns is [12], which is achievable if the number of columns is congruent to 0, 1, or 3 modulo 4, while the best achievable delay when the number of columns is congruent to [13]. 2 modulo 4 is In this paper studying the tradeoffs among transceiver signal linearization, rate, and decoding delay, we show that maximum CODs that achieve the lower bound on delay can rate achieve transceiver linearization if and only if the number of columns is odd. When the number of columns is even, a maximum rate COD must achieve at least twice the lower bound on delay (which is the best possible delay when the number of columns is 2 modulo 4 [13]) in order to achieve this transceiver signal linearization. II. PRELIMINARIES For any COD with or columns, and for any fixed , , Liang showed that the orthogonality constraint through equivalence implies that it is possible to transform operations (e.g., rearrange order of rows/columns, negate rows/ columns, conjugate/negate all instances of a given variable) so
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that the following submatrix , which contains all instances of the variable (up to conjugation and sign), appears within the or rows, respectively, [2] top
If is of maximum rate, Liang proved several properties conand submatrics [2]. For example, he showed cerning the columns, then has order that if a maximum rate has . If has columns, then is either or . In this case, we will assume that is ; all proofs can be altered slightly for the alternative. He submatrices have no zero entries [2]. also showed that these Let any (possibly noncontiguous) 2 2 orthogonal submatrix of a COD that is isomorphic under equivalence operations to Alamouti’s original COD (see (1)) be called an Alamouti 2 2. We say that two rows of a COD share an Alamouti 2 2 over two columns if the intersection of these rows and columns forms such a 2 2 orthogonal submatrix. We now recall a simple but powerful result originally proven in Part 1) of Lemma 3.2 in [13]: and be distinct rows of an Result 2.1: [13] Let COD . Then, rows and share an Alamouti 2 2 over columns and if and only if and are simultaneously nonzero exactly in columns and and never simultaneously zero in any column. III. TRANSCEIVER LINEARIZATION OF MAXIMUM RATE COSTBCS In this section, we prove our main result by determining when a maximum rate COD can achieve conjugation-separation (as described above to mean that in any given row, the nonzero entries are either all conjugated or all nonconjugated). Then, the recent work by Su et al. [9] implies that such CODs can be implemented to achieve transceiver signal linearization. This linearization is beneficial for practical systems, particularly those involving iterative decoding [8], [10]. Theorem 3.1: Let be a maximum rate COD that achieves the lower bound on decoding delay. Then, is equivalent to a COD that is conjugation-separated, and, therefore, this arrangement of can achieve transceiver signal linearization. Proof: We can perform suitable equivalence operations to submatrix for variable (see Section II). The first create the rows of each contain zeros and one entry of , rows of each contain zeros and and the last one entry of . Since all instances of (up to conjugation and sign) appear within , all instances of contained in rows zeros are nonconjugated, and all instances of with contained in rows with zeros are conjugated. Now, without affecting the conjugation of the instances of , we can perform equivalence operations on to move from to displaying another submatrix for displaying [2], [12]. Every instance of (up to conjugation and sign) will submatrix [2], [12]. By known properties appear within this of (see Section II), it is possible to conjugate all instances of so that the instances of within rows with zeros are
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nonconjugated and the instances of within rows with zeros are conjugated. to display We can repeat this procedure by rearranging for each . When a specific is formed, it may be required to conjugate all instances of so that rows zeros have nonconjugated versions of and rows with zeros have conjugated versions of . Since every with instance of an arbitrary variable appears within , and since is this variable only undergoes possible conjugation when submatrix, this procedure ensures arranged to display the zeros is nonconjugated that any variable within a row with and any variable within a row with zeros is conjugated. is equivalent to a COD that achieves conjugationHence, separation and, hence, transceiver signal linearization. Theorem 3.2: Let be a maximum rate COD that achieves the lower bound on decoding delay. Then no arrangement of is conjugation-separated. Proof: We recall for clarity that a COD with columns for odd cannot achieve the decoding delay of [13], so the conditions of this theorem imply that is even. Now, assume for contradiction that is arranged to be conjugation-separated. We have previously shown [13] that it is possible to select rows of such that for each , row and share an Alamouti 2 2 over columns and . row Furthermore, rows will be distinct, while . We begin with row of the following form, and since is assumed to be conjugated-separated, we can assume that every entry represents a nonconjugated variable (positive or negative)
Then, row is selected as a row that shares an Alamouti 2 2 with row over columns and . Thus, its entries in must be conjugated to ensure the orthogocolumns and nality of the Alamouti 2 2. Then, by our assumption of conjugation-separation, the remaining nonzero entries of will also be conjugated. Thus, by Result 2.1, has the following form:
where represents any conjugated variable (positive or negative). Iterating this procedure allows us to start with the given and produce . To maintain conjugation-separation, , to form the requisite Alamouti 2 2 s between rows and and since row is assumed to be nonconjugated, all rows with odd subscript are nonconjugated and all rows with even subscript are conjugated. So, in particular, is nonconjugated and is conjugated. But this contradicts our earlier result that . Hence, this contradiction shows that a maximum rate COD with an even number of columns cannot simultaneously achieve the lower bound on decoding delay and conjugation-separation. Theorem 3.3: It is possible to construct a maximum rate COD with any even number of columns that simultaneously achieves
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 6, JUNE 2011
TABLE I SUMMARY OF RESULTS ON THE DELAY AND TRANSCEIVER SIGNAL LINEARIZATION (TSL) OF MAXIMUM RATE CODS WITH UP TO TEN COLUMNS
TABLE II GENERAL RESULTS INDICATING WHEN MAXIMUM RATE, MINIMUM DECODING DELAY CODS CAN ACHIEVE TRANSCEIVER SIGNAL LINEARIZATION (TSL)
conjugation-separation and twice the lower bound on decoding delay. Proof: This proof relies on the work of other authors. We first note that Su et al. use conjugation-separation (though not referred to as such) as a fundamental characteristic of their construction technique [5]. Their algorithm generates CODs with any even number of columns that achieve maximum rate, conjugation-separation, and twice the lower bound on decoding delay. Additionally, Liang’s algorithm [2] generates maximum rate CODs with an even number of columns that achieve twice the lower bound on decoding delay. Steps 3) and 4) of his algorithm ensure that his examples are conjugation-separated (though, again, not referred to as such). These algorithms show that it is possible for a maximum rate COD with any even number of columns to simultaneously achieve conjugation-separation and twice the lower bound on decoding delay. As no maximum rate COD can achieve a delay that is between the lower bound and twice the lower bound [12], this is the best we can do in terms of delay under these restrictions. IV. CONCLUSIONS This paper determines the conditions under which maximum rate COSTBCs can simultaneously achieve transceiver signal linearization and optimal decoding delay. A COSTBC achieves such linearization when the underlying COD is conjugationseparated [9]. For any odd number of columns, we showed that there exist maximum rate CODs that simultaneously achieve conjugation-separation and the lower bound on decoding delay. For any even number of columns, a maximum rate COD that achieves the lower bound on delay cannot achieve conjugationseparation; however, for any even number of antennas, there exists a conjugation-separated COD that achieves twice the lower bound on delay. In the case when the number of columns is 2 modulo 4, achieving twice the lower bound on decoding delay
ADAMS et al.: MAXIMUM RATE COMPLEX ORTHOGONAL SPACE-TIME BLOCK CODES
is the best possible delay, while in the case of 0 modulo 4, this is twice the best achievable delay. These results are summarized in Table I for up to ten columns, while Table II summarizes the general results. These results contribute to the understanding of the tradeoffs that must be made when considering the optimization and implementation of COSTBCs. Of course, it is only for two antennas that we can simultaneously achieve full rate, square size (meaning lowest possible delay), and transceiver signal linearization. As the number of antennas increases, there is a sacrifice in some or all of these parameters. ACKNOWLEDGMENT The authors would like to thank J. Pollack for his work on related results. They would also like to thank the anonymous reviewers for their helpful comments on preliminary drafts of this paper. REFERENCES [1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, May 1999. [2] X.-B. Liang, “Orthogonal designs with maximal rates,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2468–2503, Oct. 2003. [3] A. V. Geramita and J. Seberry, Orthogonal Designs, ser. Lecture Notes in Pure and Applied Mathematics. New York: Marcel Dekker, 1979, vol. 45. [4] X.-B. Liang and X.-G. Xia, “On the nonexistence of rate-one generalized complex orthogonal designs,” IEEE Trans. Inf. Theory, vol. 49, no. 11, pp. 2984–2989, Nov. 2003. [5] W. Su, X.-G. Xia, and K. J. R. Lui, “A systematic design of high-rate complex orthogonal space-time block codes,” IEEE Commun. Lett., vol. 8, no. 6, pp. 380–382, Jun. 2004. [6] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Aug. 1998. [7] S. Haykin and M. Moher, Modern Wireless Communication. Upper Saddle River, NJ: Prentice-Hall, 2004. [8] B. Lu and X. Wang, “Iterative receivers for multiuser space-time coding systems,” IEEE J. Sel. Areas Commun., vol. 18, no. 11, pp. 2322–2335, Nov. 2000. [9] W. Su, S. N. Batalama, and D. A. Pados, “On orthogonal space-time block codes and transceiver signal linearization,” IEEE Commun. Lett., vol. 10, no. 2, pp. 91–93, Feb. 2006. [10] S. Jayaweera and H. Poor, “A RAKE-based iterative receiver for spacetime block-coded multipath CDMA,” IEEE Trans. Signal Process., vol. 52, no. 3, pp. 796–806, Mar. 2004. [11] N. Al-Dhahir, A. Naguib, and A. Calderbank, “Finite-length MIMO decision feedback equalization for space-time block-coded signals over multipath-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 4, pp. 1176–1182, Jul. 2001.
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[12] S. Adams, N. Karst, and J. Pollack, “The minimum decoding delay of maximum rate complex orthogonal space-time block codes,” IEEE Trans. Inf. Theory, vol. 53, no. 8, pp. 2677–2684, Aug. 2007. [13] S. Adams, N. Karst, and M. K. Murugan, “The final case of the decoding delay problem for maximum rate complex orthogonal designs,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 103–112, Jan. 2010. Sarah Spence Adams (M’03) received the B.S. degree in mathematics (summa cum laude) from the University of Richmond, VA, and the M.S. and Ph.D. degrees in mathematics from Cornell University, Ithaca, NY, where she was also a member of the Wireless Intelligent Systems Laboratory in the Department of Electrical and Computer Engineering. Previous nonacademic experience includes appointments at the Institute for Defense Analyses, Center for Communications Research, and the National Security Agency. She is currently an Associate Professor of mathematics and electrical and computer engineering with the Franklin W. Olin College of Engineering, Needham, MA. Her research interests include discrete mathematics, algebraic error-control codes, space-time block codes, and other orthogonal matrices over finite and infinite alphabets.
Nathaniel Karst received the B.S. degree in electrical and computer engineering from the Franklin W. Olin College of Engineering, Needham, MA. He is currently a graduate student at the Center for Applied Mathematics, Cornell University, Ithaca, NY.
Mathav Kishore Murugan received the B.Tech. degree in electronics and electrical communication engineering and the M.Tech. degree in microelectronics and VLSI design from the Indian Institute of Technology, Kharagpur, India. He is currently a graduate student at the Center for Applied Mathematics, Cornell University, Ithaca, NY.
Tadeusz A. Wysocki (M’94–SM’98) received the M.Eng.Sc. degree with the highest distinction in telecommunications from the Academy of Technology and Agriculture, Bydgoszcz, Poland, in 1981, the Ph.D. degree (summa cum laude) in 1984, after three years of research in the area of modulation theory, and the D.Sc. degree (Habilitation) in telecommunications engineering 1990, both from the Warsaw University of Technology, Warsaw, Poland. He continued research in combined modulation and coding, and until the end of 1991, he was with the Academy of Technology and Agriculture. In January 1992, he moved to Australia, where he worked at different universities and research centers. He spent the 1993 at the University of Hagen, Germany, within the framework of the Alexander von Humboldt Research Fellowship, and in December 1998, he joined the University of Wollongong as an Associate Professor. In 2003, he established the Wireless Research Group within the Telecommunications and Information Technology Research Institute (TITR), which he directed until the end of 2007, when he moved to take up a Professorship of computer and electronics engineering at the Peter Kiewit Institute, University of Nebraska—Lincoln. He is the author or coauthor of seven books, over 200 research publications, and nine patents. His areas of research interest include: propagation of microwaves, diversity techniques, digital modulation and coding schemes, space-time signal processing, as well as mobile data protocols including those for ad-hoc networks and nano-networking.
